Question

The design of a physical instrument requires that there be a constant difference in length of 10 cm between an iron rod and a copper cylinder laid side by side at all temperatures. If $\alpha_{Fe} = 11 \times 10^{- 6}{^\circ}C^{- 1}$ and $\alpha_{cu} = 17 \times 10^{- 6}{^\circ}C^{- 1}$, their lengths are

• A. 28.3 cm, 18.3 cm
• B. 23.8 cm, 13.8 cm
• C. 23.9 cm, 13.9 cm
• D. 27.5 cm, 17.5 cm

Answer 1

Answer: (A)

28.3 cm, 18.3 cm

Answer 2

Since a constant difference in length of 10 cm between an iron rod and a copper cylinder is required therefore

$L_{Fe} - L_{Cu} = 10cm$ .....(i)

Or $\Delta L_{Fe} - \Delta L_{Cu} = O$ $\therefore$ $\Delta L_{Fe} = \Delta L_{Cu}$

i.e., Linear expansion of iron rod = Linear expansion of copper cylinder

⇒ $L_{Fe} \times \alpha_{Fe} \times \Delta T = L_{Cu} \times \alpha_{Cu} \times \Delta T$ ⇒ $\frac{L_{Fe}}{L_{Cu}} = \frac{\alpha_{Cu}}{\alpha_{Fe}} = \frac{17}{11}$ $\therefore$ $\frac{L_{Fe}}{L_{Cu}} = \frac{17}{11}$ .....(ii)

From (i) and (ii) $L_{Fe} = 28.3cm,L_{Cu} = 18.3cm$.